Dimensionless variables
for the channel electrode

In this appendix it is shown how the material balance equation may be reduced into a dimensionless form. A full treatment leads to the result that the dimensionless current may be expressed as a sole function of two mass transport parameters, a dimensionless time parameter and a number of dimensionless rate constants. If the Lévêque approximation is used to linearise the convective velocity profile, the two mass transport parameters reduce to a single one - the shear rate Peclet number.

A1.1 Full Treatment

A1.1.1 Steady-state transport-limited current

For a transport-limited electrolysis (for example a one electron reduction):

A + e => B(A1.1)

at a channel electrode the steady-state convective-diffusion equation is:

= 0 (A1.2)

If the dimensionless variables are introduced:

, (A1.3)

the convective-diffusion equation becomes:


Under laminar flow conditions, when a Poiseille flow profile has fully developed, vx is given by:

equation (A1.2) may be reduced into a dimensionless form:


and (A1.7)

Hence the concentration is a function of four parameters:

a = a(c,y,p1,p2)(A1.8)

The dimensionless current (Nusselt number) is given by:




c is set to zero and y disappears once the integral is evaluated, therefore the Nusselt number is purely a function of p1 and p2.

A1.1.2 Homogeneous kinetics

For an irreversible transport-limited ECE or EC2E reaction (written as a reduction):

A + e- = B
nB => C
C + e- = Products

(where n=1 defines an ECE reaction and n=2 defines an EC2E reaction), the steady-state material balance equations are:

= 0 (A1.12)
= 0 (A1.13)
= 0 (A1.14)

where kc is given by:


These can be reduced into dimensionless forms in the same way. Species A does not have any kinetic terms in its material balance equation. The equation for species B becomes:


The kinetic term is known as the dimensionless rate constant:



b = b(c,y,p1,p2,Knorm)(A1.18)

Similarly the equation for species C becomes:



c = c(c,y,p1,p2,Knorm,b) = c(c,y,p1,p2,Knorm)(A1.20)

Only species A and C are electroactive, so the dimensionless current is given by:


The first integral has been given above and if the second is treated analogously:


Where f and f' are used to represent different functions. Hence the ratio of the ECE current to the electron transfer current:


A1.1.3 Time-dependent behaviour

If the dimensionless lengths are substituted into the time-dependent mass transport equation:
A dimensionless time, t, may be defined to absorb the leading term:
where t is given by:

A1.2 Lévêque Approximation

A1.2.1 Steady-state transport-limited current

If the Lévêque approximation is made:


the dimensionless mass transport equation reduces to:

where Ps is the shear rate Peclet number:


a = a(c,y,Ps)(A1.31)

and Nu simplifies to a unique function of the shear Peclet number.

A1.2.2 Treatment without axial diffusion

If we redefine the dimensionless variable in the y co-ordinate:


the mass transport equation becomes:


which simplifies to:



p = Ps-2/3(A1.35)

If axial diffusion can be ignored, the equation reduces to:


and the dependence on p is lost. The Nusselt number is a constant (from the integral) multiplied by Ps1/3.


The constant may be found by solving equation (A1.37) analytically, giving the classical Levich equation1 :



1 V.G. Levich, Physicochemical Hydrodynamics, Prentice-Hall, Engelwood Cliffs, New Jersey, (1962), p112.